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Norm Levenberg

Publications and source records attributed to Norm Levenberg.

15 recordsLinked to original sources
arXiv2026

Pluripotential theory on algebraic curves

In previous works, the second author defined directional Robin constants associated to a compact, nonpolar subset $K$ of an algebraic curve $A$ in $\mathbb{C}^N$ and related these to a natural class of Chebyshev constants for $K$. We define a second class of Chebyshev constants for $K$; relate these two classes; and utilize each of them to define two families of extremal-like functions which can be used to recover the Siciak-Zaharjuta extremal function for $K$.

math.CV↗
arXiv2025

Around Fekete's theorem

A classical result of Fekete gives necessary conditions on a compact set in the complex plane so that it contains infinitely many sets of conjugate algebraic integers. For such sets, we demonstrate the existence of a sequence of algebraic integers such that most of their conjugates eventually lie near the set, while maintaining a bound on heights. Finally, we examine properties satisfied by the limiting distribution of a sequence of algebraic numbers.

math.CV↗
arXiv2025

Widom factors in $\mathbb C^n$

We generalize the theory of Widom factors to the $\mathbb C^n$ setting. We define Widom factors of compact subsets $K\subset \mathbb C^n$ associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets $K=K_1\times\cdots\times K_n$ of $\mathbb C^n$, where each $K_j$ is a non-polar compact subset of $\mathbb C$, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each $K_j$ is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions $w$; in particular, for the case $w\equiv 1$. Finally, we define the Mahler measure of a multivariate polynomial relative to $K\subset \mathbb C^n$ and obtain lower bounds for this quantity on product sets.

math.CV↗
arXiv2025

Equidistribution of the conjugates of algebraic units

We prove an equidistribution result for the zeros of polynomials with integer coefficients and simple zeros. Specifically, we show that the normalized zero measures associated with a sequence of such polynomials, having small height relative to a certain compact set in the complex plane, converge to a canonical measure on the set. In particular, this result gives an equidistribution result for the conjugates of algebraic units, in the spirit of Bilu's work. Our approach involves lifting these polynomials to polynomial mappings in two variables and proving an equidistribution result for the normalized zero measures in this setting.

math.CV↗
arXiv2022

Random Polynomials in Several Complex Variables

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$ with i.i.d. complex random variable coefficients $\{a_j\}$ where $\{p_j\}$ form an orthonormal basis for a Bernstein-Markov measure on a compact set $K\subset {\bf C}^d$. Here $m_n$ is the dimension of $\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\bf C}^d$. We consider more general bases $\{p_j\}$, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow $H_n(z):=\sum_{j=1}^{m_n} a_{nj}p_{nj}(z)$; i.e., we have an array of basis polynomials $\{p_{nj}\}$ and random coefficients $\{a_{nj}\}$. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of $\frac{1}{n}\log |H_n|$ in $L^1_{loc}({\bf C}^d)$ to the (weighted) extremal plurisubharmonic function for $K$. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.

math.CV↗
arXiv2019

$C-$Robin Functions and Applications

We continue the study in the setting of pluripotential theory arising from polynomials associated to a convex body $C$ in $({\bf R}^+)^d$. Here we discuss $C-$Robin functions and their applications. In the particular case where $C$ is a simplex in $({\bf R}^+)^2$ with vertices $(0,0),(b,0),(a,0)$, $a,b>0$, we generalize results of T. Bloom to construct families of polynomials which recover the $C-$extremal function $V_{C,K}$ of a nonpluripolar compact set $K\subset {\bf C}^d$.

math.CV↗
arXiv2019

Monge-Ampère of Pac-Man

We show that the Monge-Ampère density of the extremal function $V_P$ for a non-convex Pac-Man set $P\subset {\bf R}^2$ tends to a finite limit as we approach the vertex $p$ of $P$ linearly but with a value that may vary with the line. On the other hand, along a tangential approach to $p$ the Monge-Ampère density becomes unbounded. This partially mimics the behavior of the Monge-Ampère density of the union of two quarter disks set $S$ of Sigurdsson and Snaebjarnarson. We also recover their formula for $V_S$ by elementary methods.

math.CV↗
arXiv2018

A global domination principle for P-pluripotential theory

We prove a global domination principle in the setting of P-pluripotential theory. This has many applications including a general product property for P-extremal functions. The key ingredient is the proof of the existence of a strictly plurisubharmonic P-potential.

math.CV↗
arXiv2015

An Orthogonality Property of the Legendre Polynomials

We give a remarkable additional orthogonality property of the classical Legendre polynomials on the real interval $[-1,1]$: polynomials up to degree $n$ from this family are mutually orthogonal under the arcsine measure weighted by the degree-$n$ normalized Christoffel function.

math.CA↗
arXiv2012

Pluripotential energy and large deviation

We generalize our previous results relating pluripotential energy with the electrostatic energy of a measure given by Berman, Boucksom, Guedj and Zeriahi. As a consequence, we obtain a large deviation principle for a canonical sequence of probability measures on a nonpluripolar compact set K in C^n. This is a special case of a result of R. Berman. For n=1, we include a proof that uses only standard techniques of weighted potential theory.

math.CV↗
arXiv2010

Weighted Pluripotential Theory Results of Berman-Boucksom

The main goal of these notes, compiled in 2008-2009, is to present a more-or-less self-contained discussion of some of the recent results and techniques of R. Berman and S. Boucksom in the setting of weighted pluripotential theory. We include some background results on pluripotential theory and weighted pluripotential theory, although many items are stated without proof (references are provided). Although slightly dated and certainly not error-free, we hope someone finds them helpful.

math.CV↗
arXiv2010

Pluripotential Energy

For probability measures $μ$ on compact subsets of $\CC^n$ we define two functionals $J(μ)$ and $W(μ)$ modeled on discrete approximations to $μ$ and multivariate Vandermonde determinants. We show that these functionals coincide, up to a constant, with the electrostatic energy of $μ$ defined in a more general setting by Berman, Boucksom, Guedj and Zeriahi. This generalizes the classical notion of logarithmic energy of a measure in the complex plane; i.e., the case $n=1$.

math.CV↗
arXiv2007

Strong asymptotics for Christoffel functions of planar measures

We prove a version of strong asymptotics of Christoffel functions with varying weights for a general class of sets E and measures in the complex plane. This class includes all regular measures in the sense of Stahl-Totik on regular compact sets E in the plane and even allows varying weights. Our main theorems cover some known results for subsets E of the real line R; in particular, we recover information in the case of E=R with Lebesgue measure dx and weight w(x) = exp(-Q(x)) where Q(x) is a nonnegative, even degree polynomial having positive leading coefficient.

math.CV↗